The general, kubic and icosahedral harmonics

The theorems described in section 3.2 provide for the ready calculation of the r , and Fs reducible characters, generated by the transformation properties of s, p and d-atomic orbitals distributed over the vertices of the structure orbits of the various point groups, which decompose into the direct sums of irreducible components listed in Tables 3.1 to 3.4. Application of the theorems requires the identification of sufficient numbers of central harmonics to act as basis functions for the irreducible components of the regular orbits of these molecular point groups. 

The general spherical harmonics are familiar, in low order, as the mutually orthonormal angular components of valence atomic orbitals. Now, the sufficient number of these functions to provide basis functions for the regular representations of the molecular point groups, in 

Extra considerations are required to construct suitable sets of polynomials, which provide basis functions for the irreducible subspaces of the cubic and icosahedral point groups. Clearly, such a set of central functions is invariant under the point group G. Eor such a function, f, then (fgi, fg2,. .., fgn is a subspace of the central functions invariant under G. But it is not, in general, an irreducible subspace, i.e. it may contain further subspaces that transform according to different irreducible representations. 

To generate an irreducible G subspace, for particular cases, f needs to be chosen with care. In the case of the kubic harmonics, first defined by Bethe in 1929 suitable functions are the mononomials x y zP, which we identify in Elert s notation as (mnp). The kubic harmonics up to level 4 and their maps onto the irreducible representations of the cubic groups are listed in Table 3.9. 

for instance, (100) generates ((100), (010), (001)) of type Tiu under the actions of the symmetry operators of the Oh point group. But, if it is required that the second Tiu is required to be orthogonal to the first one, with respect to integration over the unit sphere, then it is necessary to modify this second function with a Gram-Schmidt type transformation to obtain the distinct second set of Tiu symmetry, (5(300)-3(100), 5(030)-3(010), 5(003)-3(001)). 

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